Birch and Swinnerton-Dyer conjecture

The conjecture by Birch and Swinnerton-Dyer is one of the most important unsolved problems of modern mathematics and makes statements about number theory on elliptic curves .

formulation

The conjecture says something about the rank of elliptic curves . Elliptic curves are given by equations of the third degree in x and second degree in y, the “discriminant” of which D does not vanish. Rational points can be added on these curves using a "secant tangent method" investigated by Henri Poincaré in 1901 so that the result is again a rational point on the curve. This "addition" is defined geometrically as follows: a straight line is drawn through two rational points P and Q. If the straight line intersects the curve at a third point, this is reflected on the x-axis, which again provides a point on the curve, since this is symmetrical to the x-axis. The rational point of the curve thus obtained is the sum P + Q. The point at infinity (projective plane) serves as the neutral element “0”. The mirror point to P on the curve is its inverse. In the event that the straight line through P, Q does not have a third point of intersection on the curve, the point at infinity is used for this and the addition is: P + 0 = P. One can also form P + P by taking the intersection of the tangent in P as the second point in the addition construction. This construction is based on the fact that elliptic curves have Riemann surfaces in the shape of a torus (gender 1), are geometrical lattices and are therefore additive groups, which is also transferred to their behavior in rational numbers or finite fields. The existence of such a strange type of addition is also exploited in the so-called “elliptic curve” primary tests and “public key” encryption methods in cryptography . For this one needs curves with as many rational points as possible and takes advantage of the difficulty of finding the starting data for the additive generation of large rational points of the curve. See also Elliptic Curve Cryptosystems .

If you add a rational starting point P ₀ to yourself, you get a sequence of points:

${\displaystyle P_{1}=P_{0}+P_{0}}$

${\displaystyle P_{2}=P_{0}+P_{0}+P_{0}}$

${\displaystyle P_{3}=P_{0}+P_{0}+P_{0}+P_{0}}$

and so on.

Now two cases can arise - of course also on the same curve at different rational points:

1. One moves in a circle, i.e. H. any P _n is again identical to the starting point. In this case the points form a finite group. The corresponding points are called torsion points and the associated group is called a torsion group.
2. You keep coming to new points, all of which are on the curve. In this case the group would be isomorphic to the r-fold product of the whole numbers, depending on how many starting points P _{0 are} necessary to generate the rational points in this way. The number of these starting points is called the “ rank ” r of the curve.

In their conjecture, Bryan Birch and Peter Swinnerton-Dyer give a method for determining the rank of the elliptic curve from the equation. It results from the consideration of the L-function L (E, s), which is dependent on the examined elliptical curve E and a complex variable s. The L function is defined in the same way as the Riemann zeta function, but we now start from the prime number side - i.e. from the Euler product - and additionally encode the number of solutions of the elliptic curve modulo a prime number p in the series:

${\displaystyle L(E,s)=\prod _{p}{\frac {1}{(1-a_{p}p^{-s}+p^{(1-2s)})}}}$

with the number of solutions mod p . L (E, s) has the form of a correct zeta function series (as the sum over the natural numbers), it converges for real parts of s ≥3 / 2. One can now investigate whether it can be analytically continued throughout s, whether it satisfies a functional equation, where its zeros are, etc. As with the Riemann zeta function for the prime numbers, L (E, s) gives information about the asymptotic distribution of the Solutions (mod p, for large p). Birch and Swinnerton-Dyer examined the solutions with the computer in the 1960s and formulated their famous conjecture for the asymptotic distribution of the number N (p) of points on E over finite fields F (p), i.e. mod p: ${\displaystyle N(p)=\left(p-a_{p}\right)}$

Plot of the logarithm of for the elliptical curve on the vertical axis (blue color), where M runs through the first million prime numbers. It is plotted on the horizontal axis log (log ( M )), so that the BSD conjecture predicts an approximation to the straight line drawn in red (gradient equals the rank of the curve, here 1) ${\displaystyle \prod _{p\leq M}{\frac {N_{p}}{p}}}$ ${\displaystyle y^{2}=x^{3}-5x}$

${\displaystyle \prod _{p<M}{\frac {N(p)}{p}}\approx \log(M)^{r}}$ For ${\displaystyle M\rightarrow \infty .}$

It connects a product of local densities (the individual finite fields have a maximum of p elements) via the prime numbers with the asymptotic logarithmic distribution (with an exponent r, since here r “natural numbers” are available). Translated into the language of the zeta function L (E, s), it means that the order of the zero of L (E, s) at the point s = 1 - if the function has one there - equals the rank r of the group of rational points is. To do this, of course, it has to be proven that L can be continued analytically up to s = 1, so that L can be expanded into a Taylor series there. There is also a more detailed version that relates the coefficient of the Taylor expansion at the point s = 1 to arithmetic objects such as the order of the Tate-Shafarevich group, "local factors", the real period of the curve and the order of the torsion groups.

Some further theorems of number theory follow from the conjecture of Birch-Swinnerton-Dyer, for example the problem of determining congruent numbers from Jerrold Tunnell .

status

The assumption has so far only been proven in special cases:

1. In 1976, John Coates and Andrew Wiles proved that if E is an elliptic curve with “complex multiplication” and L (E, 1) is not 0, E has only a finite number of rational points. They proved this for imaginary square fields K - this is where the factor for the “complex multiplication” comes from - with class number 1; Nicole Arthaud (Arthaud-Kuhman) extended this to all imaginary square number fields.
2. In 1983 Benedict Gross and Don Zagier showed that if a modular elliptic curve has a first order zero at s = 1, then there is a rational point of infinite order.
3. In 1990 Victor Kolyvagin showed that for a modular elliptic curve for which L (E, 1) has a first-order zero at s = 1, the rank r = 1. He also showed for modular curves that r = 0 if L has no zero there.
4. In 1991 Karl Rubin showed that for elliptic curves E with complex multiplication with elements from an imaginary-square number field K, as well as with non-vanishing L-series at s = 1, the "p-part" of the Tate-Shafarevich group is that from Birch -Swinnerton-Dyer conjecture has the following order, for all prime numbers p> 7.
5. In 1999 Andrew Wiles , Christophe Breuil , Brian Conrad , Fred Diamond and Richard Taylor showed that all elliptic curves over the rational numbers are modular ( Taniyama-Shimura conjecture ), so that the results of Kolyvagin and Rubin for all elliptic curves over the rational numbers Numbers apply. ${\displaystyle \mathbb {Q} }$
6. In 2010 Manjul Bhargava and Arul Shankar showed that a positive measure of the elliptic curve over the rational numbers has rank 0 and fulfills the conjecture of Birch and Swinnerton-Dyer. In 2014, Bhargava, Christopher Skinner, and Wei Zhang showed that this is the case for the majority (over 66 percent) of elliptical curves.

For curves with groups of rank r> 1 nothing has been proven, but there are strong numerical arguments for the correctness of the conjecture.

The proof of the still open conjecture by Birch and Swinnerton-Dyer was added to their list of Millennium Problems by the Clay Mathematics Institute .

literature

• Peter Meier, Jörn Steuding and Rasa Steuding: Elliptical curves and a bold conjecture in the spectrum of science Dossier: “The greatest riddles of mathematics” (6/2009), ISBN 978-3-941205-34-5 , pages 40–47.
• Jürg Kramer The presumption by Birch and Swinnerton-Dyer , Elements of Mathematics, Volume 57, 2002, pp. 115–120, here online
• John Coates : The conjecture of Birch and Swinnerton-Dyer, in: John Forbes Nash jr., Michael Th. Rassias (eds.), Open problems in mathematics, Springer 2016, pp. 207–224

Generally in connection with elliptic curves over the rational numbers:

• Serge Lang: Fascination Mathematics , Vieweg 1989 (popular)
• ders .: Elliptic curves - diophantine analysis , Springer 1978
• Neil Koblitz: Introduction to elliptic curves and modular forms 1984, Springer
• Husemoller: Elliptic curves , Springer 1987
• Silverman: The arithmetic of elliptic curves , 1986, Springer
• Silverman, Tate: Rational points on elliptic curves 1992, Springer
• Knapp: Elliptic curves , Princeton 1992
• Avner Ash, Robert Gross: Elliptic Tales: Curves, Counting, and Number Theory , Princeton University Press 2012, ISBN 0691151199 .

Individual evidence

1. ↑ The discriminant is proportional to the product of the squares of the three root differences. If two roots of the cubic equation are the same, D. vanishes. These so-called “singular points”, at which the partial derivatives both vanish, are to be avoided. They have the shape of a node (two tangents in one point) or a point (double tangent in one point) on the x-axis. In the "normal case" the curve consists of a single curve with only one zero point ("closed" at infinity) or of two curves with an additional curve closed at the finite point with two real zeros.
2. ↑ The procedure was already known to Isaac Newton . Poincaré's treatment also showed gaps; B. he did not prove the group structure. Norbert Schappacher Développement de la loi de groupe sur une cubique , Séminaire de théorie des nombres de Paris 1988–1989, Birkhäuser, 1990, pp. 158-184
3. ↑ According to Faltings / Mordell's theorem, there are only finitely many rational points for curves with gender greater than 1
4. ↑ where the product contains a prime number p which the discriminant does not divide, a so-called “ good prime number ”. If it did this (“bad prime”), E would be the elliptic curve over the associated finite field “singular” and the procedure is then more complicated.
5. ↑ that is the group of equivalence classes of "homogeneous spaces" of the group of E over local bodies. Little is known about these groups; one does not even know whether they are finite for all elliptic curves.
6. ↑ modular means that the number of solutions mod p also results from the Fourier coefficients of a modular form , or rather, that a modular form can be formed with these numbers of solutions alone. Modular elliptic curves are also called "Weil curves".
7. ↑ To prove the Fermat conjecture, Wiles and Taylor had already proven this for special (semi-stable) elliptic curves
8. ↑ Bhargava, Shankar Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank 0 , 2010, Arxiv
9. ↑ Bhargava, Skinner, Zhang, A majority of elliptic curves over Q satisfy the Birch and Swinnerton-Dyer conjecture , Arxiv 2014

Web links

• Official problem description in the context of the Millennium Problems (PDF) by Andrew Wiles (140 kB)
• Lecture by Jürg Kramer
• Rubin, Silverberg: Ranks of Elliptic Curves , Bulletin AMS, 2002